Chapter 3

Graph each system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. \(3 x+y=3\) \(6 x+2 y=6\)

Graph each system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the given function for this region. $$ \begin{array}{l}{2 x+3 y \geq 6} \\ {3 x-2 y \geq-4} \\ {5 x+y \geq 15} \\\ {f(x, y)=x+3 y}\end{array} $$

Find the coordinates of the vertices of the figure formed by each system of inequalities. $$ \begin{array}{l}{x+y \leq 9} \\ {x-2 y \leq 12} \\ {y \leq 2 x+3}\end{array} $$

Solve each system of equations by using elimination. \(6 d+3 f=12\) \(2 d=8-f\)

Graph each system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. \(y-x=5\) \(2 y-2 x=8\)

Jonathan and members of his Spanish Club are going to Costa Rica. He purchases 10 traveler's checks in denominations of \(\$ 20, \$ 50,\) and \(\$ 100,\) totaling \(\$ 370 .\) He has twice as many \(\$ 20\) checks as \(\$ 50\) checks. How many of each denomination of traveler's checks does he have?

Graph each system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the given function for this region. $$ \begin{array}{l}{2 x+2 y \geq 4} \\ {2 y \geq 3 x-6} \\ {4 y \leq x+8} \\\ {f(x, y)=3 y+x}\end{array} $$

GEOMETRY Find the area of the region defined by the system of inequalities \(y+x \leq 3, y-x \leq 3,\) and \(y \geq-1\)

All 28 members in Crestview High School's Ski Club went on a one-day ski trip. Members can rent skis for \(\$ 16\) per day or snowboards for \(\$ 19\) per day. The club paid a total of \(\$ 478\) for rental equipment. Write a system of equations that represents the number of members who rented the two types of equipment.

Graph each system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. \(4 x-2 y=6\) \(6 x-3 y=9\)

Solve each system of equations. \(6 x+2 y+4 z=2\) \(3 x+4 y-8 z=-3\) \(-3 x-6 y+12 z=5\)

Graph each system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the given function for this region. $$ \begin{array}{l}{x \geq 0} \\ {y \geq 0} \\ {x+2 y \leq 6} \\ {2 y-x \leq 2} \\ {x+y \leq 5} \\ {f(x, y)=3 x-5 y}\end{array} $$

GEOMETRY Find the area of the region defined by the system of inequalities \(x \geq-3, y+x \leq 8,\) and \(y-x \geq-2\)

All 28 members in Crestview High School's Ski Club went on a one-day ski trip. Members can rent skis for \(\$ 16\) per day or snowboards for \(\$ 19\) per day. The club paid a total of \(\$ 478\) for rental equipment. How many members rented skis and how many rented snowboards?

The sides of an angle are parts of two lines whose equations are \(2 y+3 x=-7\) and \(3 y-2 x=9\) . The angle's vertex is the point where the two sides meet. Find the coordinates of the vertex of the angle.

Solve each system of equations. \(r+s+t=5\) \(2 r-7 s-3 t=13\) \(\frac{1}{2} r-\frac{1}{3} s+\frac{2}{3} t=-1\)

Graph each system of inequalities. Name the coordinates of the vertices of the feasible region. Find the maximum and minimum values of the given function for this region. $$ \begin{array}{l}{x \geq 2} \\ {y \geq 1} \\ {x-2 y \geq-4} \\ {x+y \leq 8} \\\ {2 x-y \leq 7} \\ {f(x, y)=x-4 y}\end{array} $$

Beatriz is checking a shipment of technology equipment that contains laser printers that cost \(\$ 700\) each and color monitors that cost \(\$ 200\) each. She counts 30 boxes on the loading dock. The invoice states that the order totals \(\$ 15,000\) . Write a system of two equations that represents the number of each item.

The graphs of \(y-2 x=1,4 x+y=7,\) and \(2 y-x=-4\) contain the sides of a triangle. Find the coordinates of the vertices of the triangle.

Solve each system of equations. \(2 a-b+3 c=-7\) \(4 a+5 b+c=29\) \(a-\frac{2 b}{3}+\frac{c}{4}=-10\)

RESEARCH Use the Internet or other reference to find an industry that uses linear programming. Describe the restrictions or constraints of the problem and explain how linear programming is used to help solve the problem.

Write an example of a system of three equations in three variables that has (-3, 5, 2) as a solution. Show that the ordered triple satisfies all three equations.

Solve each system of equations by using either substitution or elimination. \(3 p-6 q=6\) \(2 p-4 q=4\)

Compare and contrast solving a system of two equations in two variables to solving a system of equations of three equations in three variables.

Solve each system of equations by using either substitution or elimination. \(10 m-9 n=15\) \(5 m-4 n=10\)

Melissa is solving the system of equations \(r+2 s+t=3\), \(r+2 s+t=3\), \(2 r+4 s+2 t=6,\) and \(3 r+6 s+3 t=12 .\) Is she correct? Explain.

Solve each system of equations by using either substitution or elimination. \(3 c-7 d=-3\) \(2 c+6 d=-34\)

Use the table showing state populations. \(\begin{array}{|c|c|c|c|c|}\hline 1 & {\text { California }} & {25,484,000} & {567,000} \\ \hline 2 & {\text { Texas }} & {22,118,000} & {447,000} \\\ \hline 3 & {\text { New York }} & {19,190,000} & {70,000} \\ \hline 4 & {\text { Florida }} & {17,019,000} & {304,000} \\ \hline 5 & {\text { lllinois }} & {12,653,000} & {80,000} \\ \hline\end{array}\) Write equations that represent the populations of Florida and New York \(x\) years after 2003. Assume that both states continue to gain the same number of residents every year. Let \(y\) equal the population.

The general form of an equation for a parabola is \(y=a x^{2}+b x+c,\) where \((x, y)\) is a point on the parabola. If three points on the parabola are \((0,3),(-1,4),\) and \((2,9),\) determine the values of \(a, b, c .\) Write the equation of the parabola.

Solve each system of inequalities by graphing. $$ \begin{array}{l}{y < 2 x-3} \\ {y \leq \frac{1}{2} x+1}\end{array} $$

Solve each system of equations by using either substitution or elimination. \(6 g-8 h=50\) \(6 h=22-4 g\)

Solve each system of inequalities by graphing. $$ \begin{array}{l}{|x| \leq 3} \\ {|y| > 1}\end{array} $$

Solve each system of equations by using either substitution or elimination. \(2 p=7+q\) \(6 p-3 q=24\)

Solve each system of inequalities by graphing. $$ \begin{array}{l}{|x+1| \leq 3} \\ {x+3 y \geq 6}\end{array} $$

Solve each system of equations by using either substitution or elimination. \(3 x=-31+2 y\) \(5 x+6 y=23\)

Solve each system of equations by graphing. \(\frac{2}{3} x+y=-3\) \(y-\frac{1}{3} x=6\)

What is the solution to the system of equations shown below? \(\left\\{\begin{array}{l}{x-y+z=0} \\ {-5 x+3 y-2 z=-1} \\ {2 x-y+4 z=11}\end{array}\right.\) F. (0, 3, 3) G. (2, 5, 3) H. no solution J. infinitely many solutions

FARMING For Exercises \(34-37\) , use the following information. Dean Stadler has 20 days in which to plant corn and soybeans. The corn can be planted at a rate of 250 acres per day and the soybeans at a rate of 200 acres per day. He has 4500 acres available for planting these two crops. Let \(c\) represent the number of acres of corn and let \(s\) represent the number of acres of soybeans. Write a system of inequalities to represent the possible ways Mr. Stadler can plant the available acres.

Solve each system of inequalities by graphing. $$ \begin{array}{l}{y \geq 2 x+1} \\ {y \leq 2 x-2} \\ {3 x+y \leq 9}\end{array} $$

Solve each system of equations by using either substitution or elimination. \(3 u+5 v=6\) \(2 u-4 v=-7\)

Solve each system of equations by graphing. \(\frac{1}{2} x-y=0\) \(\frac{1}{4} x+\frac{1}{2} y=-2\)

The Yoder Family Dairy produces at most 200 gallons of skim and whole milk each day for delivery to large bakeries and restaurants. Regular customers require at least 15 gallons of skim and 21 gallons of whole milk each day. If the profit on a gallon of skim milk is \(\$ 0.82\) and the profit on a gallon of whole milk is \(\$ 0.75,\) how many gallons of each type of milk should the dairy produce each day to maximize profits?

Solve each system of inequalities by graphing. $$ \begin{array}{l}{x-3 y > 2} \\ {2 x-y < 4} \\ {2 x+4 y \geq-7}\end{array} $$

Solve each system of equations by using either substitution or elimination. \(3 a=-3+2 b\) \(3 a+b=3\)

Solve each system of equations by graphing. \(\frac{4}{3} x+\frac{1}{5} y=3\) \(\frac{2}{3} x-\frac{3}{5} y=5\)

Solve each system of inequalities by graphing. \(y \leq x+2\) \(y \geq 7-2 x\)

FARMING For Exercises \(34-37\) , use the following information. Dean Stadler has 20 days in which to plant corn and soybeans. The corn can be planted at a rate of 250 acres per day and the soybeans at a rate of 200 acres per day. He has 4500 acres available for planting these two crops. If the profit is \(\$ 26\) per acre on corn and \(\$ 30\) per acre on soybeans, how much of each should Mr. Stadler plant? What is the maximum profit?

Solve each system of inequalities by graphing. $$ \begin{array}{l}{x \leq 1} \\ {y < 2 x+1} \\ {x+2 y \geq-3}\end{array} $$

Solve each system of equations by using either substitution or elimination. \(0.25 x+1.75 y=1.25\) \(-0.5 x+2=2.5 y\)

Graph each system of equations and describe it as consistent and independent, consistent and dependent, or inconsistent. \(1.6 y=0.4 x+1\) \(0.4 y=0.1 x+0.25\)